1. Quiz 2 solution sketches 11:00 Lecture, Version A Note for multiple-choice questions: Choose the closest answer

2. Internal Rate of Return Norah invests $1,000 today and will receive $50 per year forever, starting in 1 hour. What is the internal rate of return on this investment? -1000 + 50 + 50/IRR = 0 50/IRR = 950 IRR = 5.26%

3. Net Present Value Jacqueline has been asked to determine the net present value of a business. The business has costs of $50,000 per year forever, starting six months from now. The business receives $40,000 in profits one year from today, and increases by 8% each year forever. What is the NPV of the business if the effective annual discount rate is 12%?

4. Net Present Value PV benefits = 40000/(.12-.08) = $1,000,000 PV costs = 50000/.12*(1.12) 1/2 = $442,959 NPV = $559,041

5. Semi-annual Perpetuity Valerie will receive $20,000 per year forever. However, the first payment will be made two years from today, and payments will be made every 6 months. (In other words, she will receive $10,000 every 6 months, starting 2 years from today.) If her effective annual discount rate is 10%, what is the PV of this stream of payments?

6. Semi-annual Perpetuity 6-month rate: (1.1) 1/2 – 1 = 4.880% PV = 10,000/.0488 * 1/(1.1) 3/2 PV = $177,588

7. Less-Frequent-than-Annual Perpetuities Harrison will receive $50,000 every four years forever, starting six years from now. The effective annual interest rate is 6%. Find the total present value of all payments. 4-year rate: (1.06) 4 – 1 = 26.2477% FV 2 = 50,000/.262477 = $190,493 PV = 190,493/(1.06) 2 = $169,538

8. Equal Loan Payments Jill Birdwell will borrow $50,000 on July 1, 2014. She will make 10 equal yearly payment, on January 1 of years 2015-2024, to completely pay back the loan. How much will each payment be if the effect annual interest rate is 8%?

9. Equal Loan Payments If payments were made on July 1 of each year: 50,000 = C/.08 * [1 – 1/(1.08) 10 ] 50,000 = 6.71008 * C C = $7,451.47 Payments are actually on January 1: C’ = C/(1.08) 1/2 C’= $7,170.18

10. Equivalent Annual Cost The Lockyear Machine can be purchased today for $500. It will incur maintenance costs of $200 two years from today and $500 five years from today. The machine lasts for 10 years. If the effective annual discount rate is 12%, what is the equivalent annual costs of the machine?

11. Equivalent Annual Cost PV costs = 500 + 200/(1.12) 2 + 500/(1.12) 5 = $943.15 EAC: 943.15 = C/.12 * [1 – 1/(1.12) 10 ] 943.15 = 5.6502 * C C = $166.92

12. Profitability Index If you invest $5,000 today in a new invention, you will receive a positive cash flow of $800 in 8 months, followed by a positive cash flow of $800 every 12 months thereafter. What is the profitability index of this investment if the effective annual discount rate is 9%?

13. Profitability Index PV benefits = 800/.09 * (1.09) 1/3 PV benefits = $9147.93 P.I. = PV benefits / Today’s cost P.I. = 9147.93 / 5000 = 1.82959

14. Growing Perpetuity Petey will receive $500 today. He will receive 5% more each subsequent year. If his effective annual discount rate is 15%, what is the present value of this stream of payments? PV = 500 + 525/(.15-.05) = $5,750

15. Long-Answer: Internal Rates of Return Rachel will be opening a gold mine this year. All of the gold will be extracted in one year, and she must close it in two years to meet environmental laws. This leads to the following cash flows: She knows that she must spend $200 million today, she will earn $460 million one year from today, and she must spend $262 million two years from today.

16. Long-Answer: Internal Rates of Return

17. (b) For what annual discount rates will this project have positive net present value? You must show examples to justify your answer and get full credit. Pick a discount rate in each range: r = 0%: NPV = -200 + 460 – 262 = -2 < 0 NO r = 10%: NPV = -200 + 460/1.1 – 262/1.1 2 = 1.65 > 0 YES r = 100%: NPV = -200 + 460/2 – 262/2 2 = -35.5 < 0 NO

18. Long-Answer: Internal Rates of Return (b) Pick a discount rate in each range: r = 0%: NPV = -200 + 460 – 262 = -2<0 NO r = 10%: NPV = -200 + 460/1.1 – 262/1.1 2 = 1.65>0 YES r = 100%: NPV = -200 + 460/2 – 262/2 2 = -35.5<0 NO Positive NPV for r: 0.0381965<r<0.2618035